geneasco.py
I had been toying around with the idea, inspired by NMR, to probe "stuff" (I will expand on this) in CA in a systematic manner that can yield useful results. I was surprised there has not been much study in the dynamics of rules in general and CGoL in particular starting from random configurations, and I have devised a script that can allow both this and, in the future, probing specific pattern families.

What is geneasco.py?
It is a Python script that runs ns 16x16 soups in any rule for kmax generations, recording generation by generation their population AND their population change (the average is given by the results divided by ns, obviously).
The idea behind this is that random soups are not random in their evolution and, on top of significant background noise, can show surprisingly important trends in population and population change when averaged across enough examples.

What information this gives us: Population
I greatly suggest you run the script for CGoL with ns=2000 and kmax=2000 (should be fairly quick, less than half a minute) and plot the resulting cumulative population against generation in Excel. The resulting plot is what seems to be a simple Morse potential-like function. The parameters for that Morse potential-like function will be very interesting to understand, as it will give insight into the dynamics of the rule/soup size/soup density/etc. You will see surprisingly well-defined humps within that general function though, and I think we can safely call this fine-structure, which I think has vital information about the rule's dynamics (think X-ray crystallography in chemistry - the fine detail within the diffraction points gives info on the atom positions in the unit cell).Unexpectedly, if you run it for 5000 generations especially for loads of scans (40000), you see the population falling again after a peak at about 2100. Interestingly, if you zoom in enough with these many scans, you see a VERY well-defined fine structure!*

finestructure.JPG (102.19 KiB) Viewed 4797 times

Which brings us to this. That "fine-structure" I have found to differ extremely significantly between relatively closely-related rules. I think further studying this will be extremely interesting.

What information this gives us: Population Change
Population change converges apparently oscillating around 0 (actually, around the expected end heat of a soup).
Plotting ln(change^2)/2 vs generation shows something unique: after a given generation threshold, the behaviour shows a slow decrease with a lot of noise mostly under a maximum y=mx+c, related to the chaotic heat of the rule. However, before this generation threshold, in the case of CGoL being around 160 generations, the curve followed is not linear and very precise, behaving in an entirely different manner. The complexity of the system seems to be defined by generation 160-ish for CGoL, before which the evolution of most soups has strikingly similar features (in fact, the dip at about 28 ticks is actually where you are most likely to find B-heptominos, r-pentominos, pis etc!).

concertedevolution.JPG (59.31 KiB) Viewed 4792 times

*NOTE: I upload the raw results from the 5000-generation 40k scans as a text file.

Future work:
Discrete Fourier transforms will be HUGELY useful for the fine structure (not the Morse potential-like function which is already easily analysed). For some reason, the frequency domain always shows more processable data. I will try to find out how to do this.

I have run 120000 scans through 8000 generations in CGoL and I can confirm that, for a larger number of scans, the fine structure becomes identifiable for more generations (even down to generation 5000).

I have noticed some interesting behaviour by plotting ln((pop-minpop)+1) vs ln(changepop^2)/2, with CGoL showing areas that correspond to artificially-random patterns (i.e. initial generations of a soup) and areas that correspond to natural-like behaviour (such as 60 generations after the interaction between a Pi and a constellation).
This addresses the issue of obtaining useful "natural-like" soups for synthesis and its premise could be used in design of parents! (the check is computationally really cheap)

More things, I have found something so far unique to CGoL (vs tlife, Snowflakes, etc) which is that after the initial dip bottleneck, the population actually increases again. Also, tlife, Snowflakes and even HighLife, rules that seem very close to CGoL otherwise, do NOT have the same concerted evolution behaviour when plotting ln(changepop^2)/2 vs generation, implying that it has a less complex dynamic behaviour.

B37/S23 (DryLife) shows an increase in population after an initial dip, as expected. However, as opposed to CGoL, after the initial dip in ln(changepop^2)/2 vs generation, it shows a much earlier onset of dispersion in change in population, which keeps steadily increasing instead of decreasing.

So far, the only rule to show similar behaviour to CGoL is B37e/S23, which, beyond the collective [linear growths]*P(linear growths) after gen 5470, shows a similar dip followed by a rapid increase with loads of fine structure, although obviously reaching higher populations. A less pronounced concerted evolution region than with CGoL.

Instead of saving the results in a text file it uses overlay commands to create 2 plots that are saved in .png files.

The plots are created using a function called xyplot, so people might like to pull out that code and use it in other projects that need to display results in some sort of plot. The function is quite versatile and easy to use. For example, this piece of code:

I apologise for having ignored this project (and CGoL to some extent, actually) for the past few months.
I started my PhD about one and a half months ago and now I am pretty busy. I will attempt to devote some time to this when I have a bit more free time, if that is something I can expect from the coming years!

For those who might wish to continue in this line:
I remember I was unsure whether the discrete Fourier transform script I used actually worked properly for periodic patterns that were out-of-phase wrt the reference wave to which it was assigned, so caveat emptor and keep an eye. And yes, discrete FTs will be the way to go to get any meaningful info out of this, but the dependence on phase coherence with the script I used leads to poor results.
I haven't been able to get to grips with why ln(changepop^2)/2 vs generation shows pseudo-random behaviour ONLY after ca. 160 generations, not before. This seems related to the very nature of the CA rule but I have no idea how to approach it.

So far, the only rule to show similar behaviour to CGoL is B37e/S23, which, beyond the collective [linear growths]*P(linear growths) after gen 5470, shows a similar dip followed by a rapid increase with loads of fine structure, although obviously reaching higher populations. A less pronounced concerted evolution region than with CGoL.

Have you tried analyzing rules one isotropic transition from CGOL, like b38/s23 or b3/s238?

Have you tried analyzing rules one isotropic transition from CGOL, like b38/s23 or b3/s238?

... B37e/S23 is one isotropic transition from CGOL. Also, Rhombic isn't back per se: they're still quite busy from the looks of it.
EDIT: toroidalet has informed me that the purpose of that post was probably to ask about other Life-like rules. Apologies.

"A man said to the universe:
'Sir, I exist!'
'However,' replied the universe,
'The fact has not created in me
A sense of obligation.'" -Stephen Crane

Apologising again for my long absence, I think I have cracked the problem of the following trend mentioned in the OP:

However, before this generation threshold, in the case of CGoL being around 160 generations, the curve followed is not linear and very precise, behaving in an entirely different manner. The complexity of the system seems to be defined by generation 160-ish for CGoL, before which the evolution of most soups has strikingly similar features (in fact, the dip at about 28 ticks is actually where you are most likely to find B-heptominos, r-pentominos, pis etc!).

While I am no mathematician, I think there is a clear attractor-like behaviour for the soups. I will build on this tomorrow but there are reasons to be very excited about this.

Please recall that the dPop was highly predictable up to circa generation 120-160, after which it entered into the chaotic heat convergence. The reasons for this had long been obscure, but after modelling 12000 generations of 29% fill 7x7 soups with a huge number of scans (N = 696337) in order to observe the fine structure down to the last thousand ticks, the behaviour is much better defined than expected.

The following image is a plot of Pop vs dPop, normalised to the average soup by dividing the totals by 696337, connected by lines showing that from generation to generation it moves in a predictable manner across this 2D plane.
We can clearly see that the first 6/7 ticks are entirely unnatural and show the greatest disparities, after which all soups start converging onto a trend leading to progressively higher populations with a positive dPop until 29 live cells are reached more or less. While the curve between Pop=24 and Pop=29 may look absurd at first, I have noticed that with more and more scans it does not act like noise (going to zero everywhere) but it actually keeps the same shape. A closer inspection shows some positive and negative dPop cycles of similar length that become tighter and tighter until reaching the small cone visible at about Pop=25.2

dpopvspop.JPG (47.04 KiB) Viewed 2198 times

And if now we zoom in to the finer structure area, which, I insist, is NOT background noise, what is clear is that there are multi-generation cycles with positive or negative dPop, which gradually show smaller dPops that converge to zero but always moving from positive to negative in shorter and shorter cycles - this is only properly visible with enough scans and for long runs of >8000 ticks:

dpopvspop121.JPG (80.54 KiB) Viewed 2198 times

As we can see, the final cone is well defined and tends to 25 cells at 12000 generations, showing that whatever we can call the system to be behaving as - the internal energy of the soup, the entropy, the heat... the point is the following:

A random soup generally is NOT "real"-Life-like, i.e. it will likely be extremely unlikely to have such a soup appearing out of random evolution. After 10 ticks approximately, we have a pattern that evolves naturally and we can call active, which, depending on the initial population/density conditions, will maintain a positive dPop in a metastable manner (which I predict to never end for DryLife) until it reaches the dPop/Pop "attractor" at which it consumes eventually in a purely convergent manner in the final cone. This final cone is a purely statistical feature at which a given percentage of the soups, by generation N, have converged. At generation 12000, the extremely low dPop and stable Pop show a fairly clear convergence.